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... one is willing to consider the more complicated Powell-Sabin split, the order can be reduced further to k = d − 1 [66,68]. The two refinement patterns are shown for the two dimensional case in Figure 2.1. In this work we will consider the case of k ≥ d on barycentrically refined meshes, but the arguments apply mutatis mutandis to the Powell-Sabin split. ...
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... that though M H and M h form a nested hierarchy, this is not true forˆMforˆ forˆM H andˆMandˆ andˆM h . This two-level approach canonically extends to many levels; a hierarchy of three levels is shown in Figure 2.2. ...
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... the mesh hierarchy in Figure 2.2, we notice that the interpolation is exact along the edges of the coarse-grid macro mesh. ...
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... addition, we compare to the [P 2 ] 2 −P 0 element used in [7,25] which converges at first order only. As motivated in the introduction, since the Taylor-Hood and the [P 2 ] 2 −P 0 element do not enforce the divergence constraint exactly (see bottom right of Figure 5.2), the velocity error increases as the Reynolds number is increased. This is in contrast to the solutions obtained using the Scott-Vogelius element, which are divergence-free up to solver tolerances and exhibit Reynolds-robust errors. ...
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... one is willing to consider the more complicated Powell-Sabin split, the order can be reduced further to k = d − 1 [66,68]. The two refinement patterns are shown for the two dimensional case in Figure 2.1. In this work we will consider the case of k ≥ d on barycentrically refined meshes, but the arguments apply mutatis mutandis to the Powell-Sabin split. ...
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... that though M H and M h form a nested hierarchy, this is not true forˆMforˆ forˆM H andˆMandˆ andˆM h . This two-level approach canonically extends to many levels; a hierarchy of three levels is shown in Figure 2.2. ...
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... the mesh hierarchy in Figure 2.2, we notice that the interpolation is exact along the edges of the coarse-grid macro mesh. ...
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... addition, we compare to the [P 2 ] 2 −P 0 element used in [7,25] which converges at first order only. As motivated in the introduction, since the Taylor-Hood and the [P 2 ] 2 −P 0 element do not enforce the divergence constraint exactly (see bottom right of Figure 5.2), the velocity error increases as the Reynolds number is increased. This is in contrast to the solutions obtained using the Scott-Vogelius element, which are divergence-free up to solver tolerances and exhibit Reynolds-robust errors. ...
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Renormalization enables a systematic scale-by-scale analysis of multiscale systems. In this paper, we employ \textit{renormalization group} (RG) to the shell model of turbulence and show that the RG equation is satisfied by $ |u_n|^2 =K_\mathrm{Ko} \epsilon^{2/3} k_n^{-2/3}$ and $ \nu_n = \nu_* \sqrt{K_\mathrm{Ko}} \epsilon^{1/3} k_n^{-4/3}$, where...
Citations
... The linear systems associated with coupled discretizations such as those arising from (1.5), which are often called nonsymmetric saddle point systems, can be very difficult to solve. While significant progress has been made in recent years [2,16,19,24], solving these systems when ν is small can be slow and sometimes not completely robust. Projection and penalty methods both avoid the need to solve such linear systems, as we see below, and thus with these methods it is typically much easier to get numbers. ...
... It is now well-established that the characterization of the kernels of discretized differential operators is crucial for the design of robust multigrid schemes [7,32,33,80]; for both the MTW and AW-type elements, this is given precisely by their positions in the discrete exact complexes (3.2) and (4.11), as we now explain. We describe multigrid relaxation in the framework of subspace correction methods [88]. ...
... Building on the work of Benzi and Olshanskii [15], Schöberl [80], and Hong et al. [44,45] among others, Farrell and coauthors have successfully developed parameter-and mesh-robust preconditioners of augmented Lagrangian (AL) type, with specialized multigrid algorithms, for a host of nonlinear PDEs with saddle point structure [29,30,31,33,34,35,59,87]. We illustrate the AL method for AW elements, the application to MTW being analogous. ...
The Arnold–Winther element successfully discretizes the Hellinger–Reissner variational formulation of linear elasticity; its development was one of the key early breakthroughs of the finite element exterior calculus. Despite its great utility, it is not available in standard finite element software, because its degrees of freedom are not preserved under the standard Piola push-forward. In this work we apply the novel transformation theory recently developed by Kirby [SMAI-JCM, 4:197–224, 2018] to devise the correct map for transforming the basis on a reference cell to a generic physical triangle. This enables the use of the Arnold–Winther elements, both conforming and nonconforming, in the widely-used Firedrake finite element software, composing with its advanced symbolic code generation and geometric multigrid functionality. Similar results also enable the correct transformation of the Mardal–Tai–Winther element for incompressible fluid flow. We present numerical results for both elements, verifying the correctness of our theory.
... The resulting indefiniteness in these matrices, along with the coupling between the two unknowns in the problem, prevents the straightforward application of standard geometric or algebraic multigrid methods. Several successful preconditioners for these systems are known in the literature, built on approximate blockfactorization approaches [1,3,4] or monolithic geometric multigrid methods [5,6,7,3,8,9]. While some algebraic multigrid algorithms have been proposed in the past [10,11,12,13,14], general-purpose algebraic multigrid methods that are successful for the numerical solution of a wide variety of discretizations of the Stokes equations are a recent (and still rare) development [15,16,17,18]. ...
... The structured meshes are generated using the Firedrake software, while the unstructured meshes are generated using Gmsh [52]. Barycentric refinement of meshes uses the code associated with [4]. As test problems, we consider the following, with domains pictured in Figure 4. ...
In this paper, we investigate a novel monolithic algebraic multigrid solver for the discrete Stokes problem discretized with stable mixed finite elements. The algorithm is based on the use of the low-order $\pmb{\mathbb{P}}_1 \text{iso}\kern1pt\pmb{ \mathbb{P}}_2/ \mathbb{P}_1$ discretization as a preconditioner for a higher-order discretization, such as $\pmb{\mathbb{P}}_2/\mathbb{P}_1$. Smoothed aggregation algebraic multigrid is used to construct independent coarsenings of the velocity and pressure fields for the low-order discretization, resulting in a purely algebraic preconditioner for the high-order discretization (i.e., using no geometric information). Furthermore, we incorporate a novel block LU factorization technique for Vanka patches, which balances computational efficiency with lower storage requirements. The effectiveness of the new method is verified for the $\pmb{\mathbb{P}}_2/\mathbb{P}_1$ (Taylor-Hood) discretization in two and three dimensions on both structured and unstructured meshes. Similarly, the approach is shown to be effective when applied to the $\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$ (Scott-Vogelius) discretization on 2D barycentrically refined meshes. This novel monolithic algebraic multigrid solver not only meets but frequently surpasses the performance of inexact Uzawa preconditioners, demonstrating the versatility and robust performance across a diverse spectrum of problem sets, even where inexact Uzawa preconditioners struggle to converge.
... The linear systems associated with coupled discretizations such as those arising from (1.5), which are often called nonsymmetric saddle point systems, can be very difficult to solve. While significant progress has been made in recent years [2,6,14,10], solving these systems when ν is small can be slow and sometimes not completely robust. Projection and penalty methods both 2 avoid the need to solve such linear systems, as we see below, and thus with these methods it is typically much easier to 'get numbers'. ...
We study continuous data assimilation (CDA) applied to projection and penalty methods for the Navier-Stokes (NS) equations. Penalty and projection methods are more efficient than consistent NS discretizations, however are less accurate due to modeling error (penalty) and splitting error (projection). We show analytically and numerically that with measurement data and properly chosen parameters, CDA can effectively remove these splitting and modeling errors and provide long time optimally accurate solutions.
... One defect of the Alfeld split is that it does not commute with a multigrid subdivision. However, an optimal-order convergent, non-nested multigrid method has been developed in [FMSW21]. ...
We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in 3D with exact divergence constraints. More precisely, we compare the standard Scott-Vogelius elements of higher polynomial degree and low order methods on split meshes, the Alfeld and the Worsey-Farin split. The main tool is a counting strategy to express the degrees of freedom for given polynomial degree and given split in terms of few mesh quantities, for which bounds and asymptotic behavior under mesh refinement is investigated. Furthermore, this is used to obtain insights on potential precursor spaces in full de Rham complexes for finite element methods on the Worsey-Farin split.
... For the B-E formulation (2.1.1) Hu et al. [63] show that both a Lagrange multiplier and an augmented Lagrangian term lead to a pointwise preservation of [14,41] and the second term to enforce the divergence constraint ∇ · B = 0. ...
... The key components for a robust multigrid method for the SPSD augmented Lagrangian terms are a parameter-robust relaxation method, that efficiently damps error modes in the kernel of the singular operators, and a kernel-preserving prolongation operator, as revealed in the seminal work of Schöberl [98]. The non-symmetric terms are more troublesome, but numerical results have shown [41,42] that subspace correction methods can still perform well for the Navier-Stokes equations at high Reynolds numbers. ...
... This corresponds to the standard Newton linearisation of the Navier-Stokes equations with an augmented Lagrangian term, plus the linearisation of the Lorentz force D. We follow the approach of [58,42,41] to solve this system. The first idea is to use the augmented Lagrangian term −γ∇∇ · u to approximate the inner Schur complement of the hydrodynamic block by choosing a large γ, e.g., γ ≈ 10 4 . ...
The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In the first part of this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretisation of the $\mathbf{B}$-$\mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. Our approach relies on specialised parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. In the second part, we focus on incompressible, resistive Hall MHD models and derive structure-preserving finite element methods for these equations. We present a variational formulation of Hall MHD that enforces the magnetic Gauss's law precisely (up to solver tolerances) and prove the well-posedness of a Picard linearisation. For the transient problem, we present time discretisations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. In the third part, we investigate anisothermal MHD models. We start by performing a bifurcation analysis for a magnetic Rayleigh--B\'enard problem at a high coupling number $S=1{,}000$ by choosing the Rayleigh number in the range between 0 and $100{,}000$ as the bifurcation parameter. We study the effect of the coupling number on the bifurcation diagram and outline how we create initial guesses to obtain complex solution patterns and disconnected branches for high coupling numbers.
... The continuous Stokes complexes in 3D [15] is given by ...
In recent years a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust, i.e. the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions, but remain poorly understood in three dimensions. In this work we state two conjectures on this subject. The first is that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for velocity degree $k \ge 4$; the best result available in the literature is for $k \ge 6$. The second is that there exists a stable space decomposition of the kernel of the divergence for $k \ge 5$. We present numerical evidence supporting our conjectures.
... Divergence-free finite element discretizations also allow for an easier characterization of the kernel of the discretized grad-div term. This characterization has applications in preconditioners for systems arising in incompressible fluid flow [26,27,32,40,48]. ...
... However, barycentrically refined meshes can be difficult to align with jumps in the material distribution that solves the infinite-dimensional problem, which can lead to poorly resolved solutions. Moreover, barycentrically refined meshes complicate the generation of a mesh hierarchy for robust multigrid cycles [26]. In contrast, there exist low-order divergence-free DG finite element methods that are inf-sup stable on general meshes. ...
... To our knowledge, the stability results mentioned so far for the Scott-Vogelius element have only be obtained for homogeneous Dirichlet or Neumann boundary conditions on Γ. However, there is numerical evidence that the stability and optimal accuracy of this finite element method also occur on barycentric refinements of shape-regular meshes when mixed boundary conditions are imposed [29,16]. ...
We introduce a pure--stress formulation of the elasticity eigenvalue problem with mixed boundary conditions. We propose an H(div)-based discontinuous Galerkin method that imposes strongly the symmetry of the stress for the discretization of the eigenproblem. Under appropriate assumptions on the mesh and the degree of polynomial approximation, we demonstrate the spectral correctness of the discrete scheme and derive optimal rates of convergence for eigenvalues and eigenfunctions. Finally, we provide numerical examples in two and three dimensions.
... Before achieving the following presented results, we tried to use the Newton's iteration (5.1) or (5.2) with the Stokes initial data but both did not converge at higher Reynolds numbers 1 ν =: Re 1000. Hence, to solve this Re-restraint problem for the standard Newton's iteration method and avoid all factitious terms in the numerical schemes, we decide to follow the so-called continuation method, which was successfully applied in [7,13,14,32]. The specific approach is that the problem is first solved for Re = 100, then Re = 400, 1000, 1800, 2500, 3200, 5000, and then in steps of 2500 until Re = 15000 under the coarse mesh and until Re = 20000 under the refined mesh, with the solution for the previous value of Re used as initial guess for the next; the Stokes equations are solved to provide the initial guess used at Re = 100. ...
In this work, we develop a high-order pressure-robust method for the rotation form of the incompressible Navier-Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H(div)-conforming velocity reconstruction operator. In order to match the rotation form and ease error analysis, a skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proved to achieve the pressure-independent velocity errors. Optimal convergence rates of H1 , L2-error for the velocity and L2-error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and remarkable performance of the proposed method.
